Francesco Rossi

Benedetto Piccoli, Francesco Rossi

Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that $L^1$ spaces are not natural for such equations, since we lose uniqueness of the solution.

Giacomo Albi, Mattia Bongini, Francesco Rossi et al.

We provide a mean-field description for a leader-follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions for the leader-follower dynamics, under suitable assumptions. We then establish, for an appropriate choice of the initial datum, the equivalence of the system with a PDE-ODE system, that consists of a continuity equation over the state space and an ODE for the transition from leader to follower or vice versa. We further introduce a stochastic process approximating the PDE, together with a jump process that models the switch between the two populations. Using a propagation of chaos argument, we show that the particle system generated by these two processes converges in probability to a solution of the PDE-ODE system. Finally, several numerical simulations of social interactions dynamics modeled by our system are discussed.

Sivanand Achanta, Albert Antony, Ladan Golipour et al.

Recent advances in text-to-speech (TTS) synthesis, such as Tacotron and WaveRNN, have made it possible to construct a fully neural network based TTS system, by coupling the two components together. Such a system is conceptually simple as it only takes grapheme or phoneme input, uses Mel-spectrogram as an intermediate feature, and directly generates speech samples. The system achieves quality equal or close to natural speech. However, the high computational cost of the system and issues with robustness have limited their usage in real-world speech synthesis applications and products. In this paper, we present key modeling improvements and optimization strategies that enable deploying these models, not only on GPU servers, but also on mobile devices. The proposed system can generate high-quality 24 kHz speech at 5x faster than real time on server and 3x faster than real time on mobile devices.